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Title: Boundedness, blowup and critical mass phenomenon in competing chemotaxis
Authors: Jin, HY
Wang, ZA 
Issue Date: 2016
Publisher: Academic Press
Source: Journal of differential equations, 2016, v. 260, no. 1, p. 162-196 How to cite?
Journal: Journal of differential equations 
Abstract: We consider the following attraction repulsion Keller-Segel system: u(t) = Delta u - del . (chi u del upsilon)n+ del . (xi u del omega), x is an element of Omega, t >0, upsilon(t) = Delta u + alpha u -beta upsilon, x is an element of Omega, t >0, 0= Delta w + gamma u - delta w, x is an element of Omega, t >0, u(x, 0) = u(0)(x), upsilon(x, 0) = upsilon 0(x), x is an element of Omega, with homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-2 with smooth boundary. The system models the chemotactic interactions between one species (denoted by u) and two competing chemicals (denoted by u and w), which has important applications in Alzheimer's disease. Here all parameters chi, xi, alpha, beta, gamma and delta are positive. By constructing a Lyapunov functional, we establish the global existence of uniformly-in-time bounded classical solutions with large initial data if the repulsion dominates or cancels attraction (i.e., 4 xi gamma >= alpha chi). If the attraction dominates (i.e., xi gamma < alpha chi), a critical mass phenomenon is found. Specifically speaking, we find a critical mass m* = 4 pi/alpha chi-xi gamma. such that the solution exists globally with uniform-in-time bound if M < m* and blows up if M > m* and M is not an element of {4 pi m : m E N+} where N+ denotes the set of positive integers and M = iQuoclx the initial cell mass.
ISSN: 0022-0396
DOI: 10.1016/j.jde.2015.08.040
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